A382993 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where A(n,k) = -(1/n) * Sum_{d|n} phi(n/d) * (-k)^d.

1, 2, 0, 3, -1, 1, 4, -3, 4, 0, 5, -6, 11, -4, 1, 6, -10, 24, -21, 8, 0, 7, -15, 45, -66, 51, -10, 1, 8, -21, 76, -160, 208, -119, 20, 0, 9, -28, 119, -330, 629, -676, 315, -34, 1, 10, -36, 176, -609, 1560, -2590, 2344, -831, 60, 0, 11, -45, 249, -1036, 3367, -7750, 11165, -8226, 2195, -100, 1

Offset: 1

Author

Seiichi Manyama, Apr 11 2025

Keywords

• sign
• tabl

Examples

			Square array begins:
  1,   2,    3,    4,     5,     6,      7, ...
  0,  -1,   -3,   -6,   -10,   -15,    -21, ...
  1,   4,   11,   24,    45,    76,    119, ...
  0,  -4,  -21,  -66,  -160,  -330,   -609, ...
  1,   8,   51,  208,   629,  1560,   3367, ...
  0, -10, -119, -676, -2590, -7750, -19565, ...
  1,  20,  315, 2344, 11165, 39996, 117655, ...
		

Crossrefs

Columns k=1..5 give A000035, (-1)^(n+1) * A074763(n), A343465, A343466, A343467.

Main diagonal gives A382998.

Cf. A000010, A075195, A286957, A382994, A383011.

Programs

PARI

a(n, k) = -sumdiv(n, d, eulerphi(n/d)*(-k)^d)/n;

Formula

A(n,k) = (1/n) * A382994(n,k).

A(n,k) = -(1/n) * Sum_{j=1..n} (-k)^gcd(n,j).

G.f. of column k: Sum_{j>=1} phi(j) * log(1 + k*x^j) / j.

Product_{n>=1} 1/(1 - x^n)^A(n,k) = Product_{n>=1} (1 + k*x^n).